Proof Unit Day 1 [Chapter 2.5] (You will need to keep this with you at all times)
What is a proof?
1) A logical argument showing that the truth of a hypothesis guarantees the truth of the conclusion.
2) A logical argument in which each statement you make is backed by a statement that is accepted as true.
There are several different kinds of proof: paragraph, informal, flow, & two-column proofs.
Prove that if 3x – 2 = x – 8, then x = -3.
Given: 3x – 2 = x – 8 (the hypothesis)
Prove: x = -3 (the conclusion)
Proof:
Statements Reasons
a. 3x – 2 = x – 8 a. Given
b. 2x – 2 = -8 b. Subtraction Property of Equality
c. 2x = -6 c. Addition Property of Equality
d. x = -3 d. Division Property of Equality
The first item in the ______column is usually the statement following Given: On occasion, there is good reason to put the Given statement later in the proof. The corresponding reason listed next to the Given statement will be “______”.
The last item in the ______column is always the statement following Prove: However, the corresponding reason is never “Prove”! Prove is not ever a valid reason! If you ever put Prove in the reasons column – it will be wrong.
You will use “______”, ______, ______, ______, and ______in the ______column to justify each step. When you use one of these, its conclusion must match the statement on the left.
Every postulate and theorem is a ______statement
Every definition is a ______statement
Algebraic Proofs
Properties of Equality
/Symbols
/Examples
Addition
/ If a = b, then a + c = b + c / If x = -3, then x + 4 = -3 + 4Subtraction
/ If a = b, then a – c = b – c / If r + 1 =7,then r + 1 – 1 = 7 – 1
Multiplication / If a = b, then ac = bc / If = 8, then (2) = 8(2)
Division / If a = 2 and c ≠ 0, then = / If 6 = 3t, then =
Reflexive / a = a / 15 = 15
Symmetric / If a = b, then b = a / If n = 2, then 2 = n
Transitive / If a = b, and b = c, then a = c / If y = 32 and 32 = 9,
then y = 9
Substitution / If a = b, then b can be substituted for a in any expression. / If x = 7, then 2x = 2(7)
Prove that if 2(a + 1) = -6, then a = -4.
Given: 2(a + 1) = -6 (the hypothesis)
Prove: a = -4 (the conclusion)
Proof:
Statements Reasons
a. 2(a + 1) = -6 a.
b. 2a + 2 = -6 b. ______
c. 2a = -8 c. ______
d. a = -4 d. ______
Your Turn Practice: Fill in the blanks
Prove that if 4(g – 3) = -20, then g = -2.
Given: 4(g – 3) = -20
Prove: g = -2
Proof:
Statements Reasons
a. 4(g – 3) = -20 a.
b. 4g – 12 = -20 b.
c. 4g = -8 c.
d. g = -2 d.
Prove that if 2(x – 3) = 8, then x = 7.
Given: 2(x – 3) = 8
Prove: x = 7
Proof:
______
a. a.
b. b.
c. c.
d. d.
Prove that if 5 + x = 2x, then x = 5.
Given: 5 + x = 2x
Prove: x = 5
Proof:
______
1. Given: 2x + 4 = 8
Prove: x = 2
Statements Reasons
a. 2x + 4 = 8 a.
b. 2x = 4 b.
c. x = 2 c.
2. Given: 6x + 14 = 7x + 12
Prove: x = 2
Statements Reasons
a. 6x + 14 = 7x + 12 a.
b. 14 = x + 12 b.
c. 2 = x c.
d. x = 2 d.
3. Given: 12x – 8 = 14x + 6
Prove: x = -7
Statements Reasons
a. 12x – 8 = 14x + 6 a.
b. 12x = 14x + 14 b.
c. -2x = 14 c.
d. x = -7 d.
4. Given: x = -9
Prove: x = -15
Statements Reasons
a. x = -9 a.
b. 3x = -45 b.
c. x = -15 c.
5. Given: 7x – 3 = 5x + 3
Prove: x = 3
______
6. Given: (x + 10) = -3
Prove: x = -25
a. (x + 10) = -3 a.
b. x + 2 = -3 b.
c. x = -5 c.
d. x = -25 d.
7. Given: 5 = -3x – 4
Prove: x = -3
8. Given: – 3 = 10
Prove: x = 78
Statements Reasons
a. – 3 = 10 a.
b. = 13 b.
c. x = 78 c.